Question

(Trigonometry) A ramp is needed to allow vehicles to climb a 2-foot wall. The angle of elevation for the vehicles to safely go up must be 30° or less, and the longest ramp available is 5 feet long. Can this ramp be used safely? Provide us with the angle x to the nearest 10th and a yes or no.

Answer 1

The ramp is safe.

The unknown angle measure is, (
`23.6`)

Explanation:

Assume that the ground makes a (90) degree angle with the wall. This means that the formed triangle is a right triangle. Thus, one can use the right angle trigonometric ratios. These ratios are the following,

`sin(\theta)=(opposite)/(hypotenuse)\\\\cos(\theta)=(adjacent)/(hypotenuse)\\\\tan(\theta)=(opposite)/(adjacent)`

The hypotenuse is the side opposite the right angle of a right triangle. This side's name doesn't change. However, the (opposite) and (adjacent) sides change names based on the angle one is using to describe them in the ratio.

In this case, one is given the side opposite to an angle, as well as the hypotenuse. However, one isn't given the measure of the angle. Thus, one should use the ratio of sine (sin) to find the unknown side. Let theta (
`\theta`) represent the measure of the unknown angle.

`sin(\theta)=(opposite)/(hypotenuse)`

Substitute,

`sin(\theta)=(2)/(5)`

Inverse operations,

`\theta=sin^-1((2)/(5))`

Simplify,

`\theta=23.578178`

Since the value of (
`\theta`) is less than (30) degrees, the ramp can be used safely.